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Type:

Package

Title:

Aggregated Functional Data Calibration using Splines and Wavelets

Version:

1.0.0

Description:

Implements methods for calibrating an aggregated functional data model using wavelets or splines. Each aggregated curve is modeled as a linear combination of component functions and known weights. The component functions are estimated using wavelets or splines. The package is based on dos Santos Sousa (2024) <doi:10.1515/mcma-2023-2016> and Saraiva and Dias (2009) <doi:10.47749/T/UNICAMP.2009.471073>.

URL:

https://github.com/VitorRibasP/FunctionalCalibration

Imports:

wavethresh

License:

GPL-3

Encoding:

UTF-8

LazyData:

true

RoxygenNote:

7.3.2

Depends:

R (≥ 3.5)

NeedsCompilation:

Packaged:

2025-06-17 22:34:32 UTC; vitor

Author:

Vitor Perrone

[aut, cre], Alex Sousa

[aut]

Maintainer:

Vitor Perrone <vitor.perrone10@gmail.com>

Repository:

CRAN

Date/Publication:

2025-06-19 07:30:05 UTC

Bayesian Shrinkage

Description

A Bayesian shrinkage method applied to empirical coefficients d, aiming to denoise them.

The shrinkage function is defined as:

\delta(d) = \displaystyle \frac{(1 - p) \int_{\mathbb{R}} (\sigma u + d) \, g(\sigma u + d; \tau) \, \phi(u) \, du}{\frac{p}{\sigma} \phi\left( \frac{d}{\sigma} \right) + (1 - p) \int_{\mathbb{R}} g(\sigma u + d; \tau) \, \phi(u) \, du}

where \phi(x) is the probability density function of the standard normal distribution, and g(\theta; \tau) is the logistic density function.

Usage

Bayesian_Shrinkage(d, tau, p, sigma, MC = FALSE)

Arguments

d

Numeric value of the empirical coefficient to be denoised.

tau

Numeric value of \tau.

p

Numeric value of p.

sigma

Numeric value of \sigma.

MC

A logical evaluating to TRUE or FALSE indicating if the integrals will be approximated using Monte Carlo.

Value

A numeric value representing the result of the Bayesian shrinkage applied to the empirical coefficient d.

Logistic Density

Description

Computes the function:

g(\theta; \tau) = \displaystyle \frac{\exp\left(-\frac{\theta}{\tau}\right)}{\tau \left(1 + \exp\left(-\frac{\theta}{\tau}\right)\right)^2}

Usage

Logistic_Density(theta, tau)

Arguments

theta

Numeric value of \theta.

tau

Numeric value of \tau.

Value

A numeric value representing the result of the function g(\theta; \tau) for the specified inputs.

Functional Data Calibration with Splines

Description

This function performs functional calibration based on the following model:

A_i(x_m) = \displaystyle \sum_{l=1}^{L} y_{il} \alpha_l(x_m) + e_i(x_m), \quad i = 1,...,I, \quad m = 1,...,M = 2^J

where the functions \alpha_l(x) are estimated using spline basis functions.

In matrix notation, the model is represented as:

A = \alpha y + e

Usage

functional_calibration_splines(data, weights, x, n_functions = 10)

Arguments

data

A matrix M x I where each column represents one sample of the aggregated function — the matrix A in the model.

weights

A matrix L x I representing the weight values associated with each sample — the matrix y in the model.

x

A numeric vector of values at which the function is evaluated.

n_functions

Number of spline basis functions to be used for estimating \alpha_l(x).

Value

The function returns a list containing two objects.

alpha: A matrix with the estimated functional coefficients \alpha.
Plots: A list of plot objects, each representing the corresponding function \alpha_l(x).

References

Saraiva, M. A., & Dias, R. (2009). Analise não-parametrica de dados funcionais: uma aplicação a quimiometria (Doctoral dissertation, Master’s thesis, Universidade Estadual de Campinas, Campinas).

Examples

functional_calibration_splines(simulated_data$data, simulated_data$weights, simulated_data$x)
functional_calibration_splines(simulated_data$data, simulated_data$weights, simulated_data$x, 12)

Functional Data Calibration with Wavelets

Description

This function performs functional calibration based on the following model:

A_i(x_m) = \displaystyle \sum_{l=1}^{L} y_{il} \alpha_l(x_m) + e_i(x_m), \quad i = 1,...,I, \quad m = 1,...,M = 2^J

where the functions \alpha_l(x) are estimated using wavelet decomposition.

In matrix notation, the model is represented as:

A = \alpha y + e

Usage

functional_calibration_wavelets(
  data,
  weights,
  wavelet = "DaubExPhase",
  method = "bayesian",
  tau = 1,
  p = NULL,
  sigma = NULL,
  MC = FALSE,
  type = "soft",
  singular = FALSE,
  x = NULL
)

Arguments

data

A matrix M x I where each column represents one sample of the aggregated function — the matrix A in the model.

weights

A matrix L x I representing the weight values associated with each sample — the matrix y in the model.

wavelet

A string indicating the wavelet family to be used in the Discrete Wavelet Transform (DWT).

method

A string specifying the shrinkage method applied to the empirical wavelet coefficients. Options are: "bayesian", "universal", "sure", "probability", or "cv".

tau

A numeric value for the \tau parameter in the Bayesian shrinkage. If NULL, it is estimated from the data.

p

A numeric value for the p parameter in the Bayesian shrinkage. If NULL, it is estimated from the data.

sigma

A numeric value for the \sigma parameter in the Bayesian shrinkage. If NULL, it is estimated from the data.

MC

A logical evaluating to TRUE or FALSE indicating if the integrals in the Bayesian shrinkage are approximated using Monte Carlo simulation.

type

A string indicating whether the thresholding should be "soft" or "hard" (applies only when the method is not "bayesian").

singular

A logical evaluating to TRUE or FALSE indicating if it adds a small constant (1e-10) to the diagonal of yy^T to stabilize the matrix inversion.

x

A numeric vector of values at which the function is evaluated. If NULL, the default is the sequence 1:nrow(data).

Value

The function returns a list containing two objects:

alpha: A matrix with the estimated functional coefficients \alpha.
Plots: A list of plot objects, each representing the corresponding function \alpha_l(x).

References

dos Santos Sousa, A. R. (2024). A wavelet-based method in aggregated functional data analysis. Monte Carlo Methods and Applications, 30(1), 19-30.

Examples

functional_calibration_wavelets(simulated_data$data, simulated_data$weights)
functional_calibration_wavelets(simulated_data$data, simulated_data$weights,
                                tau = 5, p = 0.95, sigma = 0.1, x = simulated_data$x)
functional_calibration_wavelets(simulated_data$data, simulated_data$weights,
                                method = "universal")

Aggregated Curve Plot

Description

Generates the plot of the aggregated curve based on the functional coefficients and their corresponding weights. The aggregated curve is computed as:

A(x) = \displaystyle \sum_{l=1}^{L} y_l \alpha_l(x)

Usage

plot_aggregated_curve(alpha, weights, title = NULL, x = NULL)

Arguments

alpha

A numeric matrix where each column represents the values of a function \alpha_l(x) evaluated at each point in x.

weights

A numeric vector with the weight values corresponding to each function \alpha_l(x).

title

A string specifying the title of the plot.

x

A numeric vector of values at which the function is evaluated. If NULL, the default is the sequence 1:nrow(alpha).

Value

The function returns the plot of the aggregated function.

Examples

plot_aggregated_curve(simulated_data$alphas, c(0.7, 0.3))
plot_aggregated_curve(simulated_data$alphas, c(0.7, 0.3),
                      "Aggregated Curve Example", simulated_data$x)

Simulated Data

Description

This is a simulated dataset designed to illustrate the functionalities of the package. It contains 100 samples of aggregated data generated from two functions, \alpha_1(x) and \alpha_2(x), with added Gaussian noise N(0, 0.1).

The functions used in the simulation are:

\alpha_1(x) = \sin(5x) e^{-x^2} \quad \alpha_2(x) = \begin{cases} -2, & x < 0 \\ 0, & 0 \leq x < 1.5 \\ 3, & x \geq 1.5 \end{cases}

The simulations were performed over an equally spaced grid of 1024 points in the interval [-1, 2]. These functions were linearly combined using random concentrations to generate the samples, with the addition of Gaussian noise.

Usage

simulated_data

Format

An object of class list of length 4.

Value

data: A data frame with 1024 rows and 100 columns.
Each column represents one sample of the aggregated functions with Gaussian noise N(0, 0.1).
weigths: A data frame with 2 rows and 100 columns.
Each column contains the random concentrations used to aggregate the two functions in each sample.
x: A numeric vector of length 1024.
The grid of x-values used in the simulation, equally spaced from -1 to 2.
alphas: A data frame with 1024 rows and 2 columns.
The true values of the functions \alpha_1(x) and \alpha_2(x) evaluated over the x grid.

Weight Estimation

Description

Estimates the weights associated with the functional coefficients \alpha_l(x) using the using Ordinary Least Squares.

The problem can be formulated as:

A(x) = \displaystyle \sum_{l=1}^{L} y_l \alpha_l(x)

where A(x) is the aggregated function evaluated at each point x, \alpha_l(x) are the functional coefficients, and y_l are the weights to be estimated.

Usage

weight_estimation(data, alpha)

Arguments

data

A numeric vector representing one sample of the aggregated function A(x), evaluated at a grid of points x.

alpha

A numeric matrix where each column represents the values of a function \alpha_l(x) evaluated at the same grid of points as data.

Value

The function returns a vector with the estimated weights obtained using Ordinary Least Squares.

Examples

weight_estimation(simulated_data$data[,1], simulated_data$alphas)